Since I'm a patzer with a terrible end game, the best I could do was apply some silicon elbow grease. (Stockfish 1.6, which is very good in the endgame, despite not using tablebases of any kind).
It couldn't find a win for Black at all.
I think the problem with the black king trying to pick off the h-pawn is that as soon as it abandons the d-pawn, White gobbles it up, and White holds the draw. (If Black avoids trading bishops after the d-pawn falls, White can capture the black h-pawn with a bishop sac and still hold the draw. And if Black trades bishops after the d-pawn capture, then the game turns into a race to promote pawns. Although Black queens first, it's a KQP vs KQ game (White has the extra pawn), and White can hold the draw.
But you have to remember, with people playing, anything can happen.
Yeah, in general, Black should keep his pawns on the light squares, although if the game goes 1...Bc5 2.Bg3, then Black can play 2...a5 or 2...b6 and still draw.